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A103222 Real part of the totient function phi(n) for Gaussian integers. See A103223 for the imaginary part and A103224 for the norm. 4
1, 1, 2, 2, 2, 2, 6, 4, 6, 0, 10, 4, 8, 6, 4, 8, 12, 6, 18, 0, 12, 10, 22, 8, 10, 4, 18, 12, 22, 0, 30, 16, 20, 8, 12, 12, 30, 18, 16, 0, 32, 12, 42, 20, 12, 22, 46, 16, 42, 0, 24, 8, 44, 18, 20, 24, 36, 16, 58, 0, 50, 30, 36, 32, 8, 20, 66, 16, 44, 0, 70, 24, 62, 24, 20, 36, 60, 8, 78, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This definition of the totient function for Gaussian integers preserves many of the properties of the usual totient function: (1) it is multiplicative: if gcd(z1,z2)=1, then phi(z1*z2)=phi(z1)*phi(z2), (2) phi(z^2)=z*phi(z), (3) z=Sum_{d|z} phi(d) for properly selected divisors d and (4) the congruence z=1 (mod phi(z)) appears to be true only for Gaussian primes. The first negative term occurs for n=130=2*5*13, the product of the first three primes which are not Gaussian primes.

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

Eric Weisstein's World of Mathematics, Totient Function

FORMULA

Let a nonzero Gaussian integer z have the factorization u p1^e1...pn^en, where u is a unit (1, i, -1, -i), the pk are Gaussian primes in the first quadrant and the ek positive integers. Then we define phi(z) = u*product_{k=1..n} (pk-1) pk^(ek-1).

MATHEMATICA

phi[z_] := Module[{f, k, prod}, If[Abs[z]==1, z, f=FactorInteger[z, GaussianIntegers->True]; If[Abs[f[[1, 1]]]==1, k=2; prod=f[[1, 1]], k=1; prod=1]; Do[prod=prod*(f[[i, 1]]-1)f[[i, 1]]^(f[[i, 2]]-1), {i, k, Length[f]}]; prod]]; Re[Table[phi[n], {n, 100}]]

CROSSREFS

Sequence in context: A263455 A283677 A260983 * A175809 A061033 A075094

Adjacent sequences:  A103219 A103220 A103221 * A103223 A103224 A103225

KEYWORD

nice,sign

AUTHOR

T. D. Noe, Jan 26 2005

STATUS

approved

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Last modified November 22 15:27 EST 2017. Contains 295089 sequences.